Efficient risk sharing: The last frontier

When rational risk-averse agents must choose among and share monetary risks, it is known that efficient sharing is typically nonlinear, even with common beliefs. Wherever it is, the sharing rule may affect the choice, randomized choice may allow everyone to gain, and indeed a randomized choice between unacceptable risks may be acceptable. An important exception occurs if the agents' utility functions are all exponential, all logarithmic, or all the same power (HARA). Then choices should accord with a group utility function of the same form independent of the sharing rule, randomization never helps, and all efficient sharing rules are linear. This self-contained paper simplifies, refines, and completes earlier analyses, identifying all exceptions; they are the linear sharing rules that make the agents' utilities agree. Aside from HARA, this can only occur for precisely one linear sharing rule or, in periodic versions of HARA, countably many.